Sequential Convex Optimization for Detecting and Locating … · 2020. 6. 1. · 7...

Sequential Convex Optimization for Detecting and Locating Blockages in1

Water Distribution Networks2

Filippo Pecci1, Panos Parpas2, and Ivan Stoianov33

1Dept. of Civil and Environmental Engineering (InfraSense Labs), Imperial College London,4

London, UK (corresponding author). Email: [email protected]

2Dept. of Computing, Imperial College London, London, UK. Email:6

[email protected]

3Dept. of Civil and Environmental Engineering (InfraSense Labs), Imperial College London,8

London, UK. Email: [email protected]

ABSTRACT10

Unreported partially/fully closed valves or other type of pipe blockages in water distribution11

networks result in unexpected energy losses within the systems, which we also refer to as faults.12

We investigate the problem of detection and localization of such faults. We propose a novel13

optimization-based method, which relies on the solution of a non-linear inverse problem with14

`1 regularization. We develop a sequential convex optimization algorithm to solve the resulting15

non-smooth non-convex optimization problem. The proposed algorithm enables the use of non-16

smooth terms within the problem formulation, and exploits the sparse structure inherent in water17

network models. The performance of the developed method is numerically evaluated to detect and18

localize blockages in a large water distribution network using both simulated and experimental19

data. In all experiments, the sequential convex optimization algorithm converged in less than three20

seconds, suggesting that the proposed fault detection and localization method is suitable for near21

real-time implementation. Furthermore, we experimentally validate the developed method for near22

real-time fault diagnosis in a large operational water network from the UK. The method is shown23

to successfully detect and localize blockages, with real system modelling uncertainties.24

1 Pecci, January 21, 2020

INTRODUCTION25

Water distribution networks (WDNs) are faced with multiple challenges due to aging infras-26

tructure, growing water demand, and more stringent environmental standards. Advanced pressure27

control schemes are being implemented to improve WDN management and operation (Wright et al.28

2015; Pecci et al. 2019), and enable the dynamically adaptive control of both hydraulic conditions29

and network connectivity (Wright et al. 2014). These advances have resulted in an increased level30

of instrumentation (e.g. pressure control valves, isolation valves) throughout the network. In order31

to benefit from these technological innovations, accurate hydraulic models, together with monitor-32

ing and diagnostic tools, are critical to support intelligent and efficient network operation. In this33

work, we consider WDNs whose hydraulic models have been calibrated to achieve a level of accu-34

racy that is compliant with model calibration guidelines (Water Research Centre 1989). After the35

initial model calibration, monitoring and diagnostic tools are necessary to continuously validate36

and maintain the hydraulic model performance.37

This paper investigates the detection and localization of blockages, which are due to unreported38

changes to status or location of valves, following various network interventions, or other type of39

blockages (e.g. the accumulation of particles at strainers of flow meters). In the following, block-40

ages are interchangeably referred to as faults. These faults cause unexpected energy losses (local41

losses), and result in inaccurate hydraulic models, affecting WDNs operation. We formulate42

the problem of detection and localization of blockages in WDNs as a non-linear inverse problem,43

where we estimate the blockage location using measured hydraulic data - for a review on inverse44

problems, see (Aster et al. 2012). Non-linear inverse problems have been previously studied to45

address a variety of estimation problems in WDNs. Some examples include hydraulic model cali-46

bration (Kapelan 2002; Piller et al. 2012; Kang and Lansey 2011; Do et al. 2016), state estimation47

(Preis et al. 2011; Piller et al. 2014), and leak detection (Pudar and Ligett James 1992; Sopho-48

cleous et al. 2019). Previous methods to identify partially/fully closed valves include Delgado49

and Lansey (2009), Wu et al. (2012), and Do et al. (2018). Delgado and Lansey (2009) imple-50

mented a transient model to detect a closed valve in a single pipeline. In comparison, studies by51


Wu et al. (2012) and Do et al. (2018) formulated and solved an inverse problem to calibrate valve52

local loss coefficients in WDNs. In particular, Do et al. (2018) implemented a combination of53

a genetic algorithm with a Levenberg-Marquardt method to identify the location of partially/fully54

closed valves in a water network with 147 pipes, with a reported overall computational time of55

more than 4 hours. As observed by Do et al. (2018), methods implemented in previous literature56

require a significant computational effort and are not suitable for near real-time implementations57

of monitoring and diagnostic tools when large scale WDNs are considered.58

The inverse problem of detection and localization of blockages in WDNs is ill-posed, due to the59

availability of fewer measurements than possible fault locations (Do et al. 2018). A standard tech-60

nique to solve ill-posed inverse problems is based on `2 (or Tikhonov) regularization (Neumaier61

1998), which was implemented by Piller et al. (2012) to regularize a joint hydraulic and water62

quality model calibration problem. The resulting optimization problem was then solved using a63

Gauss-Newton method. In this study, we assume that most of the potential faults (blockages) are64

not present at the same time. Therefore, the vector of fault states is expected to be sparse. In com-65

parison to `2 regularization, `1 regularization techniques are more effective in promoting sparsity66

within the solution vector (Tibshirani 1996), and they are commonly employed for sparse signal67

recovery (Donoho 2006). The implementation of `1 regularization within inverse problem formu-68

lations for fault estimation has been considered in other engineering frameworks (Gorinevsky et al.69

2009). However, to the best of the authors knowledge, `1 regularization techniques has not been70

applied to fault diagnosis problems in water networks.71

In this paper, we propose a novel inverse problem formulation to detect and localize blockages72

in WDNs. The considered formulation aims to detect and localize a blockage by estimating the73

corresponding fault state, defined as deviation from the nominal state of the head difference across74

a given pipe or valve. Our problem formulation is simplified with respect to the non-linear model75

used by Wu et al. (2012, Do et al. (2018) to calibrate local loss coefficients. The objective func-76

tion to be minimized relies on the sum of squares residuals (Pudar and Ligett James 1992; Kang77

and Lansey 2011; Piller et al. 2012; Piller et al. 2014; Do et al. 2018). Moreover, we add a `178


penalty term to promote sparsity within the fault state vector. Standard gradient-based optimiza-79

tion methods implemented in previous literature (Pudar and Ligett James 1992; Kang and Lansey80

2011; Piller et al. 2012; Piller et al. 2014) are not suitable for solving the considered `1 regular-81

ized inverse problem, which result in a non-smooth non-convex optimization problem. Therefore,82

we propose a sequential convex optimization algorithm to solve the considered non-smooth non-83

convex optimization problem. The developed algorithm relies on the solution of a series of convex84

sub-problems within a trust region algorithm (Conn et al. 2000). The proposed solution algorithm85

exploits the sparse structure inherent in WDN models, which is retained by the constraints of the86

convex sub-problems. In contrast to previous literature (Kang and Lansey 2011; Piller et al. 2012;87

Piller et al. 2014), the developed algorithm does not require the computation of the full inverse of88

large matrices to evaluate the derivatives of the residual vector with respect to the unknowns.89

Benefits and limitations of the proposed method are investigated using a large operational wa-90

ter network as a case study. Firstly, we test the ability of the method to accurately detect and91

localize single and multiple faults using simulated data. Furthermore, we demonstrate and validate92

a near real-time implementation of the developed fault detection and localization method using93

experimental data from the considered case study.94

PROBLEM FORMULATION95

In the following, we formulate the problem of detection and localization of blockages in WDNs96

by using a single snapshot of network operation. For multiple demand conditions, the proposed97

fault detection and localization method is implemented sequentially to solve an inverse problem at98

each time step. We consider a WDN with np links (e.g. pipes and valves), nn demand nodes, and99

n0 known head nodes (e.g. water sources, reservoirs). Known nodal demands are defined by the100

vector d ∈Rnn (measured in m3/s). The hydraulic heads at water sources (e.g. reservoirs, network101

inlets) are known. The vector of known hydraulic heads is given by h0 ∈Rn0 (measured in meters).102

In addition, we assume that when the network model includes pumps and tanks, hydraulic heads103

at pump and tank outlets are measured. In this case, we can remove pumps and tanks from the104

network, modelling them as known hydraulic heads nodes. We denote with h ∈ Rnn the vector105


of unknown hydraulic heads (measured in meters), while the vector q ∈ Rnp represents unknown106

flow rates across network links (measured in m3/s) . In addition, frictional head losses within link107

j ∈ {1, . . . ,np} are modelled by a function φ j : R→ R defined as:108

φ j(q j) = r jq j|q j|n j−1, (1)109

Resistance coefficient r j and exponent n j take different values depending on the energy loss model110

used. When link j corresponds to a valve, we set n j = 2 and r j = 8K j/(gπ2D4j), where K j and111

D j are valve local loss coefficient and diameter, respectively (Larock et al. 1999, Section 2.2.5).112

In comparison, when link j represents a pipe, different frictional energy loss models can be used.113

Darcy-Weisbach (D-W) and Hazen-Williams (H-W) formulae are two commonly used frictional114

energy loss models (D’Ambrosio et al. 2015) - see also the Appendix. Both the D-W and H-W for-115

mulae have unbounded second order derivatives around the origin (D’Ambrosio et al. 2015). How-116

ever, their first order derivatives are continuous (Abraham and Stoianov 2016). In the numerical117

experiments presented in this manuscript, we consider the hydraulic model of a large operational118

WDN from the UK, where D-W formula is used to represent frictional energy losses. In this case,119

we have n j = 2, while r j depends implicitly on the flow q j. Here, we formulate the D-W friction120

head loss model using the explicit approximation implemented in the hydraulic modelling soft-121

ware EPANET (Rossman 2000) - see the Appendix and Simpson and Elhay (2010) for a detailed122

formulation. When other frictional energy loss formulae are used, the considered fault detection123

and localization problem results in analogous formulations to what described in the following.124

Given a link ij−→ k, let u j be the corresponding fault state (measured in meters). The energy125

conservation equation is written as126

hi−hk = φ j(q j)+u j. (2)127

In contrast to previous literature (Wu et al. 2012; Do et al. 2018), we model faults caused by128

blockages as slack variables, which appear linearly within the energy conservation equations. As129


a result, the proposed formulation has a reduced degree of non-linearity with respect to previously130

published methods (Wu et al. 2012; Do et al. 2018). When u j = 0 in (2), there is no fault at link131

j and the head difference is equal to the frictional energy loss. On the contrary, when u j 6= 0, the132

head difference does not correspond to the frictional energy losses. Hence, an anomaly is detected133

on link j. When modelling and measurement errors are present, to determine if an actual fault134

(blockage) occurred, we consider the magnitude of u j, compared to the level of uncertainty - see135

Figure 9a and the related discussion in the Case Study section.136

In order to write (2) for all links in matrix form, we introduce the edge-unknown head node137

incidence matrix A12 ∈ Rnp×nn defined as138

A12( j, i) =

1 if link j enters node i

0 if link j is not connected to i

−1 if link j leaves node i

(3)139

Analogously, we define the edge-known head node incidence matrix A10 ∈ Rnp×n0 . The energy140

conservation equations can be written in compact form as:141

φ(q)+A12h+A10h0 +u = 0 (4)142

where vector u ∈ Rnp represents the unknown fault states. Furthermore, the conservation of mass143

is formulated as:144

AT12q−d = 0 (5)145

Let x = [qT hT ]T ∈ Rnp+nn be the vector of hydraulic states. Then, the network conservation laws146

can be written as:147

f(x,u) :=

φ(q)+A12h+A10h0 +uA12T q−d

= 0. (6)148We observe that function f(·) is continuously differentiable (Abraham and Stoianov 2016) and we149


have150

∂ f(x,u)∂x

=

G(q) A12A12T 0

(7)151where G(q) denotes the Jacobian matrix of function φ(·). The matrix in (7) has a saddle-point152

structure, so it is non-singular if and only if Ker(A12T )∩Ker(G(q)) = {0} (Benzi et al. 2005,153

Theorem 3.2). Such condition holds if and only if the water network model does not include loops154

with all zero flows (Nielsen 1989; Elhay et al. 2014; Abraham and Stoianov 2016). When this is155

not verified, the links involved in the loop can all be removed from the network model.156

In this manuscript, we assume that there exists a vector of flows satisfying the mass conser-157

vation laws (5). Moreover, we assume that matrix A12 ∈ Rnp×nn is full column rank. Under such158

assumptions, given a vector of fault states, there exists a unique vector of hydraulic states such that159

equation (6) is satisfied (Collins and Cooper 1978; D’Ambrosio et al. 2015) - for a detailed proof160

see Appendix. Therefore, the implicit function theorem guarantees that we can define a continu-161

ously differentiable function c : Rnp →Rnp+nn such that f(c(u),u) = 0. Function c(u) is evaluated162

by solving equation (6) for fixed u. This can be achieved using standard algorithms for hydraulic163

analysis (Todini and Pilati 1988), which are implemented in various commercial or open source164

software packages (Rossman 2000). In this work, we use the null space algorithm proposed by165

(Abraham and Stoianov 2016), a computationally efficient implementation of the Newton method166

tailored for hydraulic analysis.167

Assume that hydraulic head measurements ĥ ∈Rm1 and flow measurements q̂ ∈Rm2 are avail-168

able at different locations throughout the network. Set m := m1+m2 and let the vector of measured169

hydraulic states y ∈ Rm be y := [q̂T ĥT ]T . Define a selection matrix A ∈ Rm×Rnp+nn to select the170

hydraulic state corresponding to each measurement. Typically, water networks are characterized171

by scarce measurement data, compared to the number of unknown parameters, i.e. m

following regularized non-linear inverse problem:174

minimizeu

12‖Ac(u)−y‖22 +λ‖u‖1 (8)175

where λ ≥ 0 is a positive scalar. The `1 regularization term in (8) is expected to induce spar-176

sity within the solution vector u∗, i.e. only few components of u∗ will be non-zero (Tibshirani177

1996). For this reason, `1 regularization has been previously implemented for fault estimation in178

other engineering frameworks - as example, see (Gorinevsky et al. 2009). However, the formula-179

tion of fault detection and localization in water networks as a non-linear inverse problem with `1180

regularization has not been previously investigated. The proposed novel formulation results in a181

non-smooth non-convex optimization problem.182

SOLUTION METHOD183

In this work, we investigate mathematical optimization methods for solving Problem (8). Since184

second order derivatives of c(·) may not be well defined for some vectors u ∈ Rnp , solution algo-185

rithms for Problem (8) rely only on first order derivatives of c(·). Previously implemented methods186

for solving non-linear least-squares problems in WDNs include reduced gradient method (Pudar187

and Ligett James 1992; Kang and Lansey 2011), and Gauss-Newton based methods (Piller et al.188

2012; Piller et al. 2014). However, these methods are not suitable when non-smooth terms are189

included within the objective function. In comparison, here we investigate solution methods for190

inverse problems whose formulations include convex non-smooth functions. The proposed method191

enables a variety of objective function modelling choices, which can be be tailored for the appli-192

cations in study - some examples can be found in (Boyd and Vandenberghe 2004, Chapters 6 and193

7). Furthermore, numerical methods proposed in previous literature relied on the computation of194

the full Jacobian matrix of c(·) at each iteration - as examples, see (Kang and Lansey 2011; Piller195

et al. 2012; Piller et al. 2014). The Jacobian matrix is computed as follows. By definition of c(u),196

we have197

f(c(u),u) = 0. (9)198


Differentiating the above equation, we obtain:199

∂ f(x,u)∂x

J(u)+∂ f(x,u)

∂u= 0, (10)200

where J(u) is the Jacobian matrix of function c(·). Water distribution networks have a sparse struc-201

ture, which is retained by the network conservation laws (6) and matrices ∂ f(x,u)∂x and∂ f(x,u)

∂u - see202

(Abraham and Stoianov 2016). Nonetheless, matrix J(u) is not necessarily sparse. As a conse-203

quence, solving equation (10) requires significant computational effort when large water networks204

are considered. In this paper, we implement a solution algorithm that does not require the com-205

putation of J(u), offering a scalable method to solve the considered problem in large scale water206

networks.207

We formulate a sequential convex optimization algorithm for the solution of the non-convex208

optimization problem (8). Because of non-convexity, mathematical optimization methods to com-209

pute globally optimal solutions for Problem (8) requires the implementation of global optimization210

techniques (Tawarmalani and Sahinidis 2002). However, these algorithms require a large com-211

putational effort, and can be impractical when large water networks are considered (Pecci et al.212

2018). For this reason, we investigate the implementation of gradient-based optimization algo-213

rithms, and develop a trust-region based sequential convex optimization algorithm for solving214

Problem (8). Trust-region techniques are used to evaluate the progress achieved by the sequen-215

tial convex optimization algorithm, and modify the search space to achieve convergence to locally216

optimal solutions (Conn et al. 2000, Chapter 11).217

The considered sequential convex optimization method is an iterative algorithm, which gener-218

ates a sequence of approximate solutions to Problem (8). Given an iterate uk, a new trial iterate is219

obtained by minimizing a convex approximation of the original objective function in Problem (8):220

221

12‖Ac(u)−y‖22 +λ‖u‖1 ≈

12‖A(c(uk)+J(uk)(u−uk))−y‖22 +λ‖u‖1 (11)222

where J(uk) is the Jacobian matrix of c(·) evaluated at uk. The convex function in (11) is expected223


to be a good approximation of the original objective function in a neighbour of uk. Therefore, the224

minimization of the approximated objective function is restricted to a trust region around uk:225

minimizeu

12‖A(c(uk)+J(uk)(u−uk))−y‖22 +λ‖u‖1

subject to ‖u−uk‖∞ ≤ ∆k(12)226

where ∆k > 0 is the trust region radius. As discussed in the previous Section, matrix J(uk) is the227

solution of the adjoint equation:228∂ fk∂x

J(uk)+∂ fk∂u

= 0 (13)229

where ∂ fk∂x :=∂ f(c(uk),uk)

∂x and∂ fk∂u :=

∂ f(c(uk),uk)∂u . Since matrix J(uk) is not necessarily sparse, solv-230

ing (13) at each iteration can become impractical when large water networks are considered. In231

this manuscript, we implement an alternative approach. Firstly, we introduce auxiliary variables232

and rewrite Problem (12):233

minimizeu,x

12‖Ax−y‖22 +λ‖u‖1

subject to x = c(uk)+J(uk)(u−uk)

‖u−uk‖∞ ≤ ∆k

(14)234

As observed in the previous Section, we can assume without loss of generality that matrix ∂ fk∂x is235

non-singular. Hence, Equation (13) implies that236

J(uk) =∂ fk∂x

−1 ∂ fk∂u

(15)237

Consequently, Problem (14) is equivalent to the following convex problem:238

minimizeu,x

12‖Ax−y‖22 +λ‖u‖1

subject to∂ fk∂x

(x−xk)+∂ fk∂u

(u−uk) = 0

‖u−uk‖∞ ≤ ∆k

(16)239


where xk = c(uk). Observe that the equality constraints in Problem (16) preserve the sparsity struc-240

ture inherent in water distribution network models. Hence, Problem (16) can be efficiently solved241

by state-of-the-art convex optimization methods (Boyd and Vandenberghe 2004), or specialized242

algorithms for large scale sparse convex problems (Zheng et al. 2017).243

The implemented sequential convex optimization method is outlined in Algorithm 1 and Figure244

1b. The algorithm starts from a given initial solution u0 and computes x0 = c(u0). At iteration k,245

we solve Problem (16) and obtain trial iterates (x+,u+). Then, we compute the following ratio:246

ρk =12‖Axk−y‖

22 +λ‖uk‖1−

12‖Ac(u

+)−y‖22−λ‖u+‖112‖Axk−y‖

22 +λ‖uk‖1−

12‖Ax+−y‖

22−λ‖u+‖1

. (17)247

The value ρk compares the reduction in the original objective function achieved by (x+,u+),248

with the predicted reduction. If the ρk is large enough, iteration k is defined as successful and249

uk+1 = u+,xk+1 = c(u+), while the trust region radius is increased. If ρk is too small, we con-250

clude that the approximate model is not a good estimate of the original objective function within251

the current trust region, and the radius is reduced. Then, if a termination criterion is met, the algo-252

rithm stops. Otherwise, a new iteration is performed. We observe that each iteration of Algorithm253

1 requires solving one sparse convex problem and one system of non-linear equations (6).254

The proposed fault detection and localization method relies on the formulation of Problem (8),255

and the implementation of Algorithm 1 to solve the resulting non-convex optimization problem,256

with with u0 = 0 and ∆0 = 20. The overall process is summarized in Figure 1. In the next Section,257

the developed method is tested using a large operational network as case study.258

CASE STUDY259

The developed method is implemented to detect and localize blockages occurring in BWFLnet,260

the hydraulic model of a large operational water network from the UK operated by Bristol Water,261

Cla-Val, and Imperial College London (Wright et al. 2014) - see Figure 2. Frictional head losses262

across network pipes are modeled according to the D-W formula (Simpson and Elhay 2010). The263

network consists of 2546 nodes, 2606 pipes, 545 valves and 2 inlets (known head nodes). BWFLnet264


Algorithm 1 Trust region based sequential convex optimization

1: Initialization: η = 0.01, γ = 1.1, α = 0.5, ε1 = 10−3, ε2 = 10−42: Select u0 and ∆0;3: Set k = 0 and x0 = c(u0);4: repeat5: Trial iterate computation. Solve Problem (16), let (x+,u+) be the computed solution;6: Acceptance of trial iterate.7: If ρk ≥ η then uk+1 = u+, xk+1 = c(u+), ∆k+1 = γ∆k , and

relchange =12‖Axk−y‖

22 +λ‖uk‖1−

12‖Axk−1−y‖

22−λ‖uk−1‖1

12‖Axk−1−y‖

22 +λ‖uk−1‖1

;

else uk+1 = uk, xk+1 = xk, and ∆k+1 = α∆k;8: until relchange ≤ ε1 or min(|12‖Axk−y‖

22 +λ‖uk‖1|,∆k)≤ ε2

is equipped with a dynamically adaptive topology, where two dynamic boundary valves (DBVs)265

are open during the diurnal operation and closed at night (Wright et al. 2014), while 3 pressure266

control valves (PCVs) are optimally operated to minimize average zone pressure - see Figure 2.267

The dynamic boundary valves in BWFLnet are closed between 23:30 and 04:15, and open at 40%268

for the remaining part of network daily operation.269

PCVs and DBVs are equipped with sensors, measuring inlet and outlet hydraulic heads, and270

flow rates. Moreover, flow from both inlets is measured. Additional 27 hydraulic head mea-271

surements are available from other locations throughout BWFLnet - see Figure 2. All measured272

hydraulic data used in this manuscript have time steps of 15 minutes. We observe that the net-273

work model in study is an order of magnitude larger than those considered in previous literature,274

and it has a lower density of measurement devices - as example see (Do et al. 2018). All ex-275

periments reported in this section are conducted in MATLAB 2018b-64 bit for Linux, installed276

on a 2.50 GHz Intel Xeon(R) CPU E5-2640 0 with 12 Cores and 12 GB of RAM. The convex277

sub-problems within Algorithm 1 are reformulated as quadratic optimization problems and solved278

using GUROBI (Gurobi Optimization 2017).279

We expect the quality of the fault diagnosis to depend on sensors number and locations, and280

the value of the regularization parameter λ . The results reported in the next Sections were ob-281


tained with λ = 10−2. Future work should investigate strategies to select the best regularization282

parameter. In many applications, a practical approach to select the parameter λ is cross-validation,283

which requires solving the problem for different values of the regularization parameter, computing284

a portion of the regularization path (Friedman et al. 2010).285

In the experiments presented in sub-sections “Single fault scenarios” and “Multiple fault sce-286

narios”, we assume perfect data information, and we do not explicitly consider the effect of uncer-287

tainty. The ability of the proposed method to detect and localize blockages in presence of modelling288

uncertainties is investigated using experimental data in the third sub-section, named “Experimental289

validation”.290

Single fault scenarios291

In order to demonstrate our fault detection and localization method, we generate flow and hy-292

draulic head measurements by simulating the water network model under different fault scenarios.293

We set customer demands and hydraulic heads at the inlets to the values corresponding to 08 : 00294

am, which corresponds to peak demand.295

Firstly, we evaluate the performance of the method for detecting changes in network connec-296

tivity, i.e. fault caused by closure of links that are assumed to be open (Wu et al. 2012; Do et al.297

2018). In order to generate measurement data, we simulate the network under different fault sce-298

narios. From the set of all network links, we select 1231 links whose closure does not result in299

isolated nodes, i.e. nodes not connected to any water source - see Figure 2.300

We simulate N = 1231 different faults, one for each considered link. To simulate each fault301

(e.g. closed valve), we define a new network model, where the selected link is removed. Hydraulic302

heads and flows are computed by solving the hydraulic equations for the modified network model,303

using the null space newton method developed by (Abraham and Stoianov 2016). Let y(1), . . . ,y(N)304

be the vectors of measurements generated for each fault scenario. Models of operational water305

networks are subject to multiple sources of data and modelling errors. In these experiments, we do306

not explicitly consider uncertainty. However, when faults result in residuals that are smaller than307

the level of uncertainty experienced in WDNs models, methods based on hydraulic models can fail308


to correctly detect and localize them. In addition, faults resulting in small residuals are expected309

to have limited impact on WDN operation. In this study, we focus our analysis on fault scenarios310

that result in large residuals. We define the analysed scenarios S as follows311

S :={

i ∈ {1, . . . ,N}∣∣∣∣ 12‖Ac(0)−y(i)‖22 ≥ 1

}(18)312

For the considered case study, we have |S | = 334 - see Figure 2. For each i ∈ S , we formu-313

late Problem (8) given measurements y(i), where all network links are considered as possible fault314

locations (e.g. possible closed valves). Observe that, the formulation of of Problem (8) includes315

np = 2606 unknowns. We implement Algorithm 1 to solve Problem (8) with λ = 0.01. The inves-316

tigation of alternative strategies to select the regularization parameter for the considered problem317

is subject of future work.318

Results and discussion319

The developed sequential convex optimization method converge to a solution in all experi-320

ments, with an average computational time of 0.787 (s). This is a significant reduction compared321

to the method by Do et al. (2018), where a computational time of more than 4 hours was required322

to localize closed valves in a case study network smaller than BWFLnet. Let u∗(i) be the vector of323

parameters computed by solving Problem (8) for fault scenario i ∈S . We expect the `1 regular-324

ization to promote sparsity in u∗(i), and this is confirmed by the results reported in Figures 3b-3f.325

Since the number of possible fault locations is 2606, while the total number of available mea-326

surements is 44, we do not expect an absolute fault localization. Instead, we expect the solution327

of Problem (8) to be sparse and to provide a set of candidate fault locations. When the absolute328

value of the fault state assigned to a link is larger than 10−3 meters, we consider such link to be329

a candidate fault location (e.g. fault hotspot area). Since these experiments are conducted under330

the assumption of perfect data, such threshold is arbitrary, and it corresponds to the smallest mag-331

nitude of fault state that we aim to detect and localize. In presence of modelling uncertainty, the332

threshold can be identified using uncertainty quantification and state estimation techniques (Puig333


2010; Vrachimis et al. 2018). Therefore, for i ∈S , we define the fault location candidates C (i) as334

the index set:335

C (i) :={

j ∈ {1, . . . ,np}∣∣∣∣ |u∗(i)j | ≥ 10−3}, (19)336

Figure 3a shows an example of successful fault detection and localization. In this case, the337

solution of Problem (8) corresponds to a link sequence, which includes the actual closed link. We338

now focus on inexact and wrong diagnoses. Figure 3c shows an inexact fault localization result,339

where the fault location was not included in the set of candidates but the correct network branch340

was identified. Hence, we do not consider this experiment as unsuccessful. In comparison, the341

results reported in Figure 3e corresponds to the worst fault localization performance. The fault342

location is not included within the set of candidates and the distance to fault is equal to the 16,343

the maximum value found in this experiment. Even though the correct part of the network was344

identified, we consider this case a wrong fault localization.345

In order to evaluate the overall performance of the implemented fault detection and localization346

method, we introduce performance criteria based on graph theory. To do so, we first give some347

essential definitions.348

Definition 4.1 (Link adjacency). Two links of a WDN graph are adjacent if they have an end in349

common.350

Definition 4.2 (Line graph). (Diestel 2000, Section 1.1) The line graph of a WDN graph is a graph351

whose vertices correspond to the links of the WDN graph. Two vertices in the line graph are352

adjacent if and only if the corresponding links are adjacent.353

Definition 4.3 (Connected components). A connected component of a graph (or sub-graph) is a354

maximal set of nodes such that each pair of nodes is connected by a path. Connected components355

of a graph (or sub-graph) form a partition of the set of graph (or sub-graph) vertices.356

Firstly, we introduce a metric to assess the ability of the method to correctly identify the actual357

fault location as a fault candidate. In particular, we consider the graph distance of the set of can-358

didates to the actual fault location. Let δ (i) be the minimum length of the shortest paths between359


the fault location and the links in C (i), computed in the line graph of the graph of BWFLnet. We360

consider the following distribution function361

F1(t) :=100|S |

∣∣∣∣{i ∈S ∣∣δ (i)≤ t}∣∣∣∣. (20)362Figure 4 reports the distribution of the distance to fault in all instances. The actual fault location363

was included in the set of candidate fault locations in the majority of experiments - resulting in364

δ (i) = 0. Moreover, the distance from the set of candidates to the actual fault location is less than365

5 in roughly 80% of the experiments.366

In order to evaluate the precision of the fault localization method, we consider the cardinality of367

the set of candidate fault locations as second performance metric. Let χ(i) := |C (i)| be the number368

of candidate fault locations found in experiment i. We define the following distribution function:369

F2(t) :=100|S |

∣∣∣∣{i ∈S ∣∣χ(i)≤ t}∣∣∣∣. (21)370As shown in Figure 5, the number of selected fault candidates in each experiment is never greater371

than 31. Considering that the number of possible fault locations is np = 2606, solving Problem (8)372

has considerably reduced the set of candidate fault locations, in all experiments.373

Hydraulic models of operational water networks often include link sequences to represent long374

pipes, or to align hydraulic models and GIS information. As observed by Do et al. (2018), when a375

link sequence is considered, different combinations of head losses within individual links result in376

the same overall energy loss along the sequence. Therefore, when few measurements are available,377

all links involved in the sequence are reported as candidate fault locations - see Figure 3. As a378

consequence, we consider the fault localization method to be precise when the set of candidate fault379

locations includes the actual fault, and the sub-graph induced by the candidate links is connected.380

Hence, as third performance metric, let σ(i) be number of connected components of the sub-graph381


defined by links in C (i). We set382

F3(t) :=100|S |

∣∣∣∣{i ∈S ∣∣σ(i)≤ t}∣∣∣∣. (22)383Figure 6 shows that C (i) corresponds to a single link sequence in the majority of the experiments. In384

particular, in all successful experiments where the actual fault location is identified as a candidate,385

the sub-graph induced by the candidate links has a unique connected component, i.e. the set of386

candidate links is connected.387

Multiple fault scenarios388

We evaluate the proposed method to detect and localize multiple simultaneous faults in BWFLnet.389

In the first experiment, 3 faults were simulated at different locations - see Figure 7a. All links were390

assumed to be open in the nominal hydraulic model. Faults 1 and 3 correspond to incorrect mod-391

elling of two partially closed valves. To simulate Fault 1 we set the corresponding valve local loss392

coefficient to 1000, while Fault 3 corresponds to a valve whose local loss coefficient is set to 500.393

In addition, Fault 2 corresponds to a closed link, and it is simulated by removing such link from394

the model before performing the hydraulic simulation - analogously to the previous section. We395

formulate Problem (8) with λ = 0.01, where the measured hydraulic heads and flows are generated396

by simulating the 3 faults. Then, we implement Algorithm 1 to compute a solution and define the397

set of candidate fault locations as in (19) - see also Figure 7b.398


Similarly to the results reported in the previous section, Algorithm 1 required only 1.12 seconds400

to converge.The proposed method results in a successful fault detection and localization, with all401

3 faults being identified. Moreover, the the set of candidate fault locations has exactly 3 connected402

components - see Figures 7c - 7e. These results are not surprising, the considered links correspond403

to locations of individual faults that were correctly identified in the single-fault experiments.404

In comparison, when the set of multiple faults includes a link corresponding to an individual405

fault that was not successfully identified, the method results in incorrect localization. This is406


illustrated in the following. Analogously to the previous experiment, we simulate three faults in407

BWFLnet, namely Fault 4, 5 and 6 - see Figure 8a. Again, the nominal hydraulic model considers408

all links to be fully open. To simulate Fault 4, which corresponds to a partially closed valve,409

we set the local loss of the corresponding valve to 1000, while we simulate Fault 6 changing the410

local loss coefficient of the corresponding valve to 500. Finally, Fault 5 corresponds to a closed411

link. We formulate Problem (8) for fault detection and localization in BWFLnet, with λ = 0.01.412

Then, Algorithm 1 is implemented to compute a solution to Problem (8). The computational time413

required to converge is 0.87 seconds. The set of candidate fault locations is defined as in (19) and414

the results are reported in Figure 8. The implemented method has successfully identified Faults 4415

and 6, but it has missed Fault 5. Note that the individual failure of the link corresponding to Fault416

5 was also missed in the previous section experiments.417

Experimental validation418

We validate the proposed optimization based fault detection and localization method using419

experimental data from BWFLnet. In the following, we consider experimental data recorded in420

BWFLnet, including pressure data from 27 locations, together with inlet and outlet pressure and421

flow measurements from the PCVs and DBVs. All measured hydraulic data used in this manuscript422

have time steps of 15 minutes. A complete description of the dataset, together with the measured423

data, is available at http://dx.doi.org/10.17632/srt4vr5k38.2. The hydraulic model of424

BWFLnet was calibrated as shown in Waldron et al. (2019) using measured hydraulic data from a425

24 hr training day. Then, recorded data from a different day (24 hours) was used to validate the426

developed method. In the formulation of Problem (8), we assume that pressure and flow measure-427

ments from DBVs are not available and that their opening level is 40% for the all day. As a result,428

the closure of the DBVs at night results in unexpected energy losses, captured by the recorded data.429

These energy losses can be considered as the effect of blockages occurring at the DBV locations.430

We investigate the application of the developed method to detect and localize such blockages using431

recorded experimental data.432

At each time step, measured hydraulic heads at inlets and pressure control valves operation433


http://dx.doi.org/10.17632/srt4vr5k38.2

were modelled within the network conservation laws (6). Moreover, the total measured demand434

was distributed according to the original demand allocation determined by the network operator,435

as described in Do et al. (2018). Observe that the main source of modelling errors is represented436

by demand uncertainty and variability within the system, which might differ from the assumptions437

made when modelling BWFLnet.438


Problem (8) was formulated and solved to detect and localize blockages in BWFLnet, consid-440

ering all links as potential blockage locations. At each time step, Algorithm 1 required on average441

2.7 seconds to converge, demonstrating that the proposed method is suitable for near real-time im-442

plementation in monitoring and diagnostic tools for WDNs. In Figure 9, we report the fault state443

for the two DBVs, computed by solving Problem (8), and the measured flow, which was recorded444

but not used within the implemented fault diagnosis process. As shown in Figure 9a, estimating the445

status of DBV 1 is challenging due to model uncertainty. In fact, DBV 1 experiences a small head446

difference during diurnal operation, with a limited measured flow. Therefore, the incorrect mod-447

elling of DBV 1 does not have a significant impact on the operation of BWFLnet. In comparison,448

Algorithm 1 resulted in an accurate estimation of the status of DBV 2, with the fault state equals449

to zero during daylight (i.e. the valve is open) and non-zero at night (i.e. the valve is closed). This450

is also confirmed by the flow data shown in Figure 9b. We conclude that the implemented method451

resulted in accurate diagnosis when this can be expected.452

To further investigate the performance of the developed method to localize closed valves, we453

consider BWFLnet operation at 03:00. Both boundary valves are closed at this time of the day, but454

our model assumes that they are open at 40%. As discussed above, the effect of valve closure at455

DBV 1 is smaller than the level of modelling uncertainty experienced in BWFLnet. Therefore, the456

incorrect modelling of DBV 1 has little impact on the operation of BWFLnet, and it is not identi-457

fied by the method. In contrast, as shown in Figure 10a, the fault vector computed by Algorithm458

1 sharply identifies a set of candidate fault locations, which are shown in Figure 10b. The imple-459

mented fault detection and localization procedure has correctly identified a link sequence including460


DBV 2.461

CONCLUSION462

We have proposed a novel optimization based method for detecting and localizing blockages463

in WDNs. These faults are caused by unreported partially/fully closed valves or other pipe block-464

ages. We have formulated the considered problem as a non-linear inverse problem with `1 regu-465

larization. We have presented a sequential convex optimization algorithm for solving the resulting466

non-smooth non-convex optimization problem. The method is based on the solution of a sequence467

of sparse convex problems, which are solved efficiently using convex optimization techniques. Nu-468

merical experiments on a large operational water network show that the optimization based fault469

detection and localization performs well, achieving accurate fault diagnosis in most cases. In fact,470

in over 80% of the tested 334 fault scenarios, the actual fault location is separated from the set471

of estimated fault candidates by less than 5 links. In addition, in all experiments, the solution of472

the inverse problem via the proposed sequential convex optimization algorithm requires less than473

three seconds of computations, reducing the computational time by 4 to 5 orders of magnitude474

compared to Do et al. (2018). Furthermore, the developed method was validated for near real-475

time fault diagnosis using recorded data from a large operational water network with dynamically476

adaptive topology from the UK. The method is shown to successfully detect and localize network477

blockages using experimental data, despite unmodeled measurement noise and other modelling478

uncertainties present in the system. The results reported in this study suggest that the proposed op-479

timization based method enables the implementation of real time fault diagnosis in water networks480

with a dynamically adaptive topology. Finally, we suggest that the developed framework is ex-481

tended to other fault diagnosis problems in WDNs, including burst/leak detection and localization.482

483

DATA AVAILABILITY STATEMENT484

The hydraulic data and model of BWFLnet used during the study are available in a repository in accor-485

dance with funder data retention policies: http://dx.doi.org/10.17632/srt4vr5k38.2486

ACKNOWLEDGEMENTS487

This research was supported by EPSRC (EP/P004229/1, Dynamically Adaptive and Resilient Water488


http://dx.doi.org/10.17632/srt4vr5k38.2

Supply Networks for a Sustainable Future). We thank Cla-Val and Bristol Water for their support in the489

implementation and operation of BWFLnet.490


APPENDIX I. EXISTENCE AND UNIQUENESS OF HYDRAULIC SOLUTIONS491

The hydraulic conservation laws in (6) are:492

φ(q)+A12h+A10h0 +u = 0

A12T q−d = 0,(23)493

where function φ j(q j) represents the frictional head loss within link j, for all j ∈ {1, . . . ,np}. As494

stated in the Problem Formulation section, we assume that the set {q ∈Rnp |A12T q−d = 0} is not495

empty. Moreover, we assume that the columns in matrix A12 ∈ Rnp×nn are linearly independent.496

We have that497

φ j(q j) = r jq j|q j|n j−1, (24)498

where r j and n j depends on the type of link (i.e. valve or pipe) and the frictional head loss formula499

that is used. In particular, when link j corresponds to a valve, we have:500

φ j(q j) =8K j

gπ2D4jq j|q j| (25)501

where K j and D j are valve local loss coefficient and diameter, respectively (Larock et al. 1999,502

Section 2.2.5). In comparison, when link j corresponds to a pipe, the frictional head losses can503

be modelled by either Hazen-Williams (H-W) or Darcy-Weisbach (D-W) formulae. When H-W is504

used, we set:505

φ j(q j) =10.67L j

C1.852j D4.781j

q j|q j|0.852, (26)506

where L j, D j and C j are length, diameter and H-W coefficient of pipe j, respectively. In the case of507

D-W, the relation between r j = r j(q j) and q j is defined implicitly. Here, we implement the explicit508


approximation used in EPANET (Rossman 2000) and discussed in (Simpson and Elhay 2010):509

φ j(q j) =

128υπg

L jD4j

q j |q j| ≤ 2000 νL j8

π2gL jD5j

(∑3l=0

α lj+βlj

θ j ηl) · |q j|q j 2000 νL j < |q j| ≤ 4000 νL j

2(ln10)2

π2gL jD5j

1(lnθ j)2

· |q j|q j 4000 νL j < |q j|,

(27)510

where ν is the kinematic viscosity, while L j and D j are length and diameter of pipe j, respectively.511

Coefficients α j, β j, η j and θ j are defined as in (Simpson and Elhay 2010). In particular,512

θ j =ε j

3.7D j+

5.74(πν/4)0.9D0.9j|q j|0.9

(28)513

where ε j is the pipe roughness.514

Consider the optimization problem:515

minimizeq

E(q)

subject to A12T q−d = 0(29)516

where517

E(q) :=np

∑j=1

∫ q j0

φ(s)ds+qT (A10h0 +u) (30)518

The next result follow was proved by Collins and Cooper (1978) in a different context. For the519

sake of completeness, we derive it in a our framework.520

Theorem I.1. A pair of vectors (q,h) solves (23) if and only if q is optimal solution of Problem521

(29) and h is the corresponding Lagrangian multiplier.522

Proof. Since function φ j(·) are monotonically increasing, the objective function E(·) is strictly523

convex. Hence, Problem (29) satisfies strong duality and the associated KKT condition are a nec-524

essary and sufficient condition for optimality (Boyd and Vandenberghe 2004, Chapter 5). There-525

fore, q is an optimal solution of Problem (29), with associated Lagrangian multiplier h, if and only526


if (q,h) satisfies the following set of non-linear equations:527

∇E(q)+A12h = 0

A12T q−d = 0(31)528

The above equations yield529

φ(q)+A10h0 +u+A12h = 0

A12T q−d = 0(32)530

Since we assume rank(A12) = nn, given q∗ ∈ Rnp optimal solution of Problem (29), there exists a531

unique associated vector of Lagrangian multipliers h∗ ∈ Rnn .532

The following theorem holds.533

Theorem I.2. Problem (29) has a unique optimal solution.534

Proof. As observed in the previous Theorem, E(·) is strictly convex. Therefore, the solution of535

Problem (29) (if it exists) is unique.536

In order to prove existence of a solution to Problem (29), it suffices to show that function E(·)537

is coercive Bertsekas (1999, Proposition A.8), i.e. lim‖q‖→∞ E(q) = +∞ . Assume that link j538

corresponds to a valve. We have539

∫ q j0

φ(s)ds = r j|q j|3

3. (33)540

When link j corresponds to a pipe, it is necessary to investigate separately H-W and D-W head541

loss formulae.542

• H-W formula. In this case, we have543

∫ q j0

φ(s)ds = r j|q j|2.852

2.852(34)544


• D-W formula. Let ω j = 4000 νL j . Firstly, we observe that function φ j(·) is odd, i.e.545

−φ j(s) = φ j(−s). Hence, Definition (27) implies that546

∫ q j0

φ(s)ds =∫ |q j|

0φ(s)ds

=∫ ω j

0φ(s)ds+

∫ |q j|ω j

φ(s)ds

=∫ ω j

0φ(s)ds+

∫ |q j|ω j

2(ln10)2

π2gL jD5j

(ln(

ε j3.7D j

+5.74(πν/4)0.9D0.9j

|s j|0.9

))−2s2 ds

≥∫ ω j

0φ(s)ds+

∫ |q j|ω j

2(ln10)2

π2gL jD5j

(ln(

ε j3.7D j

+5.74(πν/4)0.9D0.9j

ω0.9j

))−2s2 ds

≥ I j +2(ln10)2

π2gL jD5j

(ln(

ε j3.7D j

+5.74(πν/4)0.9D0.9j

ω0.9j

))−2|q j|3

3

(35)

547

where548

I j =∫ ω j

0φ(s)ds− 2(ln10)

2

π2gL jD5j

(ln(

ε j3.7D j

+5.74(πν/4)0.9D0.9j

ω0.9j

))−2 ω3j3. (36)549

In conclusion, for all cases, we have550

E(q)≥np

∑j=1

(γ j +G j|q j|µ j

)+qT (A10h0 +u) (37)551

where γ j ∈ R and G j ≥ 0 and µ j > 2 are opportunely defined. Since552

lim‖q‖→+∞

np

∑j=1

(γ j +G j|q j|µ j

)+qT (A10h0 +u) = +∞, (38)553

inequality (37) implies that554

lim‖q‖→+∞

E(q) = +∞, (39)555

556


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List of Figures655

1 Flow charts describing the implemented fault detection and localization method. . . 32656

(a) Proposed fault detection and localization method. . . . . . . . . . . . . . . . 32657

(b) Sequential convex optimization algorithm. . . . . . . . . . . . . . . . . . . 32658

2 Control valves and pressure sensors locations. Locations of the faults correspond-659

ing to links in S are highlighted. . . . . . . . . . . . . . . . . . . . . . . . . . . 33660

3 Fault estimation results for three example cases: successful fault detection and lo-661

calization (Top); inexact fault localization (middle); wrong fault localization (bot-662

tom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34663

(a) Successful fault localization: fault location and candidate links are highlighted. 34664

(b) Successful fault localization: u∗ is the estimated fault state computed by665

solving Problem (8). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34666

(c) Inexact fault localization: fault location and candidate links are highlighted. . 34667

(d) Inexact fault localization: u∗ is the estimated fault state computed by solving668

Problem (8). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34669

(e) Wrong fault localization: fault location and candidate links are highlighted. . 34670

(f) Wrong fault localization: u∗ is the estimated fault state computed by solving671

Problem (8). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34672

4 Distribution of the distance of the set of candidates to the actual fault location. . . . 35673

5 Distribution of the number of candidates . . . . . . . . . . . . . . . . . . . . . . . 36674

6 Distribution of the number of connected components of the set of candidates. . . . 37675

7 Results of the first experiment with multiple faults, where candidate links are high-676

lighted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38677

(a) Location of the 3 simulated faults. . . . . . . . . . . . . . . . . . . . . . . . 38678

(b) Estimated fault state u∗ computed by solving Problem (8). . . . . . . . . . . 38679

(c) Localization performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38680

(d) Localization performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38681


(e) Localization performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38682

8 Second experiment with multiple faults, where candidate links are highlighted. . . 39683

(a) Locations of the 3 simulated faults. . . . . . . . . . . . . . . . . . . . . . . 39684

(b) Estimated fault state u∗ computed by solving Problem (8). . . . . . . . . . . 39685

(c) Localization performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . 39686

(d) Localization performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . 39687

(e) Localization performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . 39688

9 Estimated fault state u∗ at all time steps, and measured flow across the two dynamic689

boundary valves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40690

(a) DBV 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40691

(b) DBV 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40692

10 Results of the experiment at 03:00, where candidate links are highlighted. . . . . . 41693

(a) Estimated fault state u∗ computed by solving Problem (8). . . . . . . . . . . 41694

(b) Localization performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41695


1. WDN hydraulic model

2. Pressure and flow

measurements

Non-linear inverse problem formulation

Sequential convex optimization algorithm

Estimated fault states

(a) Proposed fault detection and localizationmethod.

Set and ( )= =0 0 0u 0 x c u

Compute a new iterate ( , )

solving Problem (16) (Step 5 in Algorithm 1)

+ +x u

Sufficient reduction?

Yes No

Reduce the

size of the trust

region

Update the current

solution and

increase the trust

region

Termination criteria are satisfied?

(Step 8 in Algorithm 1)

Yes

Stop

No

Step 7 in

Algorithm 1

(b) Sequential convex optimization algorithm.

Fig. 1. Flow charts describing the implemented fault detection and localization method.


Valve

Sensor

DBV 2

DBV 1

PCV 2

PCV 1

PCV 3

Fig. 2. Control valves and pressure sensors locations. Locations of the faults corresponding tolinks in S are highlighted.


Valve

Sensor

(a) Successful fault localization: faultlocation and candidate links are high-lighted.

0 1000 2000 30000

0.1

0.2

0.3Fault

(b) Successful fault localization: u∗ is theestimated fault state computed by solvingProblem (8).

Valve

Sensor

(c) Inexact fault localization: fault loca-tion and candidate links are highlighted.

0 1000 2000 30000

0.5

1

1.5

2

2.5Fault

(d) Inexact fault localization: u∗ is the es-timated fault state computed by solvingProblem (8).

Valve

Sensor

(e) Wrong fault localization: fault loca-tion and candidate links are highlighted.

0 1000 2000 30000

0.05

0.1

0.15

0.2

Fault

(f) Wrong fault localization: u∗ is the es-timated fault state computed by solvingProblem (8).

Fig. 3. Fault estimation results for three example cases: successful fault detection and localization(Top); inexact fault localization (middle); wrong fault localization (bottom).


0 5 10 150

50

100

Fig. 4. Distribution of the distance of the set of candidates to the actual fault location.


0 10 20 300

50

100

Fig. 5. Distribution of the number of candidates


0 2 40

50

100

Fig. 6. Distribution of the number of connected components of the set of candidates.


Valve

Sensor

(a) Location of the 3 simulated faults.

0 1000 2000 30000

1

2

3

4

5

Fault 1

Fault 2

Fault 3

(b) Estimated fault state u∗ computed bysolving Problem (8).

Valve

Sensor

(c) Localization performance.

Valve

Sensor

(d) Localization performance. (e) Localization performance.

Fig. 7. Results of the first experiment with multiple faults, where candidate links are highlighted.


Valve

Sensor

(a) Locations of the 3 simulated faults.

0 1000 2000 30000

0.5

1

Fault 4

Fault 5

Fault 6

(b) Estimated fault state u∗ computed bysolving Problem (8).

Valve

Sensor

(c) Localization performance.

Valve

Sensor

(d) Localization performance.

Valve

Sensor

(e) Localization performance.

Fig. 8. Second experiment with multiple faults, where candidate links are highlighted.


00:00 12:00 00:000

0.05

0.1

-0.5

0

0.5

(a) DBV 1.

00:00 12:00 00:000

0.5

1

0

2

4

(b) DBV 2.

Fig. 9. Estimated fault state u∗ at all time steps, and measured flow across the two dynamicboundary valves.


0 1000 2000 30000

0.5

1

DBV 1

DBV 2

(a) Estimated fault state u∗ computed by solvingProblem (8).

DBV 2

Valve

Sensor

(b) Localization performance.

Fig. 10. Results of the experiment at 03:00, where candidate links are highlighted.


Single fault scenariosResults and discussion

Multiple fault scenariosResults and discussion

Experimental validationResults and discussion

Sequential Convex Optimization for Detecting and Locating … · 2020. 6. 1. · 7...

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